Schedule for: 21w5121 - Quantum Field Theories and Quantum Topology Beyond Semisimplicity

This talk will be a broad survey of recent work on new q-series invariants of 3-manifolds labeled by Spin-C structures. While the original motivation for studying these invariants is rooted in topology, they exhibit a number of unexpected properties and connections to other areas of mathematics, e.g. turn out to be characters of logarithmic vertex algebras. When q tends to a root of unity, this Spin-C TQFT can be related --- via a version of the Fourier transform that physicists may naturally call "bosonization" and that exchanges 0-form and 1-form symmetries --- to the BCGP theory, due to Blanchet, Costantino, Geer and Patureau-Mirand. This relation will be discussed in a separate talk by Francesco Costantino.

In this talk I will report on a recent collaboration joint with Sergei Gukov and Pavel Putrov exploring some new relations between the non-semisimple invariants associated to the unrolled version of quantum $sl_2$ at roots of unity and the BPS series invariants. I will first consider the case of knots in the sphere and describe the conjectures on ADO polynomials and BPS series. Then I will pass to the case of invariants of closed three manifolds and describe a conjectural relation we detailed in our paper and which we proved to hold in some infinite family of cases. In the last part of the talk I will speculate on an extension of these conjectures on the level of the associated TQFTs and describe some ideas to implement this.

Arguably one of the most difficult steps in Huang's proof of the Verlinde conjecture was proving rigidity. One indicator of why this is a special (hard to verify) property is that already within the class of c_2-cofinite yet non-semisimple theories there are known counter examples to rigidity. In this talk I will present a weaker yet more tractable form of duality, which was recently shown to apply to categories of VOA modules satisfying mild assumptions. For concreteness, I will then illustrate this structure using Heisenberg and lattice VOAs (aka free bosons).

In this talk we will be focused on non-semisimple categories of modules for affine vertex (super)algebras.  If $g$ is a Lie algebra, then the affine vertex algebra $L_k(g)$ admits non-semisimple modules only beyond the category $KL_k$. But if $g$ is a Lie superalgebra, even the category $KL_k$ can contain indecomposable modules. We will first review certain general methods of constructing logarithmic (projective) modules. Then we will show how these methods can be applied on affine vertex algebras by using recent free field realizations, which are motivated by finding inverses of the Quantum Hamiltonian Reductions. We will present new realizations of logarithmic modules of nilpotent rank three for affine vertex algebra $L_k(sl_3)$ at (almost) arbitrary non-integral level $k$. (This part of the talk is a joint work with T. Creutzig and N. Genra).

The paradigm of rational (or log-rational) conformal field theory is intimately entwined with highest-weight theory for the associated vertex operator algebras. However, there are many natural examples of VOAs for which the consistency conditions of CFT require one to look beyond the highest-weight module category. I will review some recent work on examples, including the admissible-level affine VOAs of $\mathfrak{sl}_2$, and describe the central role played by the so-called relaxed highest-weight modules.

The 4D/2D duality discovered by Beem et. al associates a VOA to any 4D N=2 SCFT. The corresponding VOA has non-semisimple representations unless the 4D theory has zero Higgs branch. In this talk I overview the 4D/2D duality especially in the viewpoint of VOA theory.

I will explain how the reciprocal of the Alexander polynomial of a knot can be viewed as a universal quantum invariant associated to the Hopf algebra of regular functions on the group of affine linear transformations of the complex plane. This is consistent with the Melvin--Morton--Rozansky conjecture proven by Bar-Nathan and Garoufalidis about the relation of the colored Jones polynomials to the reciprocal of the Alexander polynomial.

$W_\infty$ algebra is a vertex operator algebra extending the Virasoro algebra by fields of spin $3,4,\dots$. It is known to admit a nice class of modules labelled by a triple of partitions. $W_\infty$ is also known to admit an alternative description in terms of the affine Yangian of $gl_1$ admitting a very concrete definition of such modules. As we will see in this talk, utilizing the charge-conjugation automorphism of $W_\infinity$ in the language of the affine Yangian leads to a new class of affine Yangian modules with non-diagonalizable action of Cartan generators and striking connection with Pandharipande-Thomas invariants.

This talk is an overview of structures and developments related to quantum groups at a root of unity and their relation to topological invariants. The subject was developing actively over the last 30 plus years, but there are still many open problems. The study of quantum groups with large center and the construction of corresponding invariants of 3-manifolds with flat connections are some of the important developing directions in this area. The other, just emerging, is the relation of these invariants with conformal field theories and VOA.

We show that pivotal module categories provide a source of symmetric Frobenius algebras. These are natural candidates for the bulk and boundary algebras in full conformal field theories for which the chiral data are encoded in a modular finite tensor category $\mathcal C$. The bulk algebra, as well as more general defect fields, can be expressed as certain coends. The structural morphisms of these coends give in particular a bulk-boundary map, whereby the whole field content of the CFT can be reconstructed from the boundary fields. Moreover, there are natural candidates for operator products of bulk (as well as defect) fields, which pass various consistency conditions, including all genus-zero constraints in Lewellen's list. In the special case of rational conformal field theories, for which $\mathcal C$ is semisimple, the conjectured expressions reproduce known results.   

The theory of quantum invariants started with the Jones polynomial and continued with the Reshetikhin-Turaev algebraic construction of invariants. In this context, the quantum group $U_q(sl(2))$ leads to the sequence of coloured Jones polynomials, and the same quantum group at roots of unity gives the coloured Alexander polynomials. We construct a unified topological model for these two sequences of quantum invariants. More specifically, we define certain homology classes given by Lagrangian submanifolds in configuration spaces. Then, we prove that the $N^{th}$ coloured Jones and $N^{th}$ coloured Alexander invariants come as different specialisations of a {\em state sum (defined over 3 variables) of Lagrangian intersections in configuration spaces.} As a particular case, we see both Jones and Alexander polynomials from the same intersection pairing in a configuration space. Secondly, we present a {\em globalised model without state sums} from recent work. We show that one can read o  both coloured Jones and coloured Alexander polynomials of colour $N$ from a {\em graded intersection between two explicit Lagrangians in a symmetric power} of the punctured disk.

In this talk we explain our recent construction of quantum invariants of smooth 4-dimensional 2-handlebodies (i.e. 4-balls with finitely many 1- and 2-handles attached)  based on  a (possibly non-semisimple) unimodular ribbon category C.   Whenever C is factorizable, the underlying invariant only depends on the boundary and signature of the 4-dimensional 2-handlebody.  On the other hand, in the example provided by the category of finite-dimensional representations of the small quantum sl2 at a root of unity q of order r ≡ 0 mod 8,  our invariant does depend on the interior of the handlebody, and it might even be useful to resolve a deep open problem in combinatorial group theory known as Andrews–Curtis conjecture. This is a joint work with Marco De Renzi.

Coulomb branches of quantum field theories are often non-compact and could lead to TQFTs that are "non-semisimple.'' In this talk, I will discuss several cases where we can gain insight into these novel TQFTs by regularizing them.

I will discuss aspects of a 3d topological QFT T(G,k) whose braided tensor category of line operators is (conjecturally) isomorphic to the derived category of modules for the quantum group U_q(g) at a 2k-th root of unity --- and whose state spaces and partition functions provide a derived generalization of associated non-semisimple TQFT's recently constructed by Costantino, Geer, and Patureau-Mirand. The field theory T(G,k) is a topological twist of a 3d N=4 Chern-Simons-matter theory, which generalizes classic Chern-Simons theory with compact group G, at level k, extending it to a non-semisimple and derived setting. More so, T(G,k) admits chiral boundary conditions supporting the Feigin-Tipunin VOA's based on g, generalizing the appearance of the chiral WZW model in Chern-Simons theory. Supersymmetric localization allows for simple calculations of some observables/invariants when G = SU(n). (Joint work w/T. Creutzig, N. Garner, and N. Geer.)

In this talk I will describe three-manifold invariants defined via surgery presentations and show that in certain cases one obtains a TQFT via the universal construction. The algebraic input is a possibly non-semisimple ribbon category together with a modified trace on a tensor ideal. We will see in examples how the invariants can pick up different properties of the ribbon category as one varies the tensor ideal. If the ribbon category is modular and the ideal is that of projective objects, the universal construction defines a TQFT on so-called admissible bordisms. If the input category is in addition semisimple, this produces the Reshetikhin-Turaev TQFT. This is joint work with J. Berger, M. De Renzi, A. Gainutdinov, N. Geer, and B. Patureau-Mirand

The famous Reshetikhin-Turaev-Witten construction of 3d Topological QFTs has as an input data a modular tensor category that is assumed to be semi-simple. In middle of 90's Lyubashenko has proposed a reasonable non-semisimple version of modular tensor categories and it was later shown that they produce mapping class group representations with new features not present in the RTW construction, e.g. infinite order of Dehn twists action. Many important examples of such categories come from two-dimensional Logarithmic Conformal Field Theories and as representation categories of small quantum groups. However, a proper TQFT construction for Lyubashenko's theory was missing. In this talk, I will show that our non-semisimple TQFT (from Ingo’s talk) provides mapping class group representations that (projectively) agree with those defined by Lyubashenko. This is a joint work with M. De Renzi, N. Geer, B. Patureau-Mirand, and I. Runkel. I will further present very recent results on actions of another fundamental group, the group of ribbon auto-equivalences of the input modular category. In the non-semisimple case, these groups are typically non-discrete, e.g. Lie groups. In an ongoing project with M. De Renzi and I. Runkel, we have shown that their action on TQFT spaces commutes with the action of the mapping class groups.

One of the most important results in vertex operator algebras is Huang's theorem that if the module category of a vertex operator algebra satisfying $C_2$-cofiniteness (plus a few relatively minor conditions) is semisimple, then it is a semisimple modular tensor category. Huang also showed that the module category of any $\mathbb{N}$-graded $C_2$-cofinite vertex operator algebra $V$ is at least a braided tensor category. In this talk, I will discuss my recent result that if this tensor category of $V$-modules is rigid, with duals given by contragredient modules, then its braiding is non-degenerate, that is, $V$-modules form a not-necessarily-semisimple modular tensor category. I will also discuss the prospects of proving rigidity for the $V$-module category in general, as well as the possibility that rigidity is preserved under vertex operator algebra constructions that are known to preserve $C_2$-cofiniteness, such as tensor products, extensions, and finite solvable orbifolds. This leads potentially to many non-semisimple modular tensor categories obtained via standard constructions applied to the triplet vertex operator algebras $\mathcal{W}(p)$, $p\in\mathbb{Z}_{\geq 2}$.

The logarithmic Kazhdan Lusztig conjecture states that certain vertex operator algebras have the same category of representations as small quantum groups - a nonsemisimple modular tensor category. In the first part of the talk, we will introduce the underlying concepts and set the conjecture in a broader context. In the second part, we report on joint work with Thomas Creutzig. We prove that the category of representations of the small quantum group uq(sl2) is characterized by its abelian structure and the existence of certain induction functors. Since these induction functors are readily available in the category of representations of the vertex operator algebras in question (from its two free field realizations), we can thereby conclude the proof of the logarithmic Kazhdan Lusztig conjecture in its smallest case.

The logarithmic Kazhdan Lusztig conjecture states that certain vertex operator algebras have the same category of representations as small quantum groups - a nonsemisimple modular tensor category. In the first part of the talk, we will introduce the underlying concepts and set the conjecture in a broader context. In the second part, we report on joint work with Thomas Creutzig. We prove that the category of representations of the small quantum group uq(sl2) is characterized by its abelian structure and the existence of certain induction functors. Since these induction functors are readily available in the category of representations of the vertex operator algebras in question (from its two free field realizations), we can thereby conclude the proof of the logarithmic Kazhdan Lusztig conjecture in its smallest case.

I'll discuss various properties of characters of several types of rational and non-rational vertex algebras. These characters in some cases agree with Schur indices of certain Argyres-Douglas theories and with Z-hat invariants of plumbed 3-manifolds. We will also discuss so called graph schemes and associated graph series. A new link between graph schemes and multiple zeta values will be presented.